EXTERNAL AUTOMORPHISMS OF ULTRAPRODUCTS OF FINITE MODELS

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1 EXTERNAL AUTOMORPHISMS OF ULTRAPRODUCTS OF FINITE MODELS PHILIPP LÜCKE AND SAHARON SHELAH Abstract. Let L be a finite first-order language and M n n < ω be a sequence of finite L-models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space ω 2 is non-empty, then there is a non-principal ultrafilter U over ω such that the corresponding ultraproduct Q U Mn is infinite and has an automorphism that is not induced by an element of Q n<ω Aut(Mn). 1. Introduction Let L be a finite first-order language and M n n < ω be a sequence of finite L-models. Given a non-principal ultrafilter U over ω, we let U M n denote the corresponding ultraproduct and [ x] U denote the U -equivalence class of an element x n<ω M n. An automorphism π of U M n is internal if there is a sequence π i Aut(M n ) n < ω such that π([ x] U ) = [ y] U holds for all x, y n<ω M n with y(n) = π n ( x(n)). We call an automorphism π of U M n external if it is not internal. If the continuum hypothesis (CH) holds and U M n is infinite, then it is a saturated model of cardinality ℵ 1 (see [1, 6.1.1] and [4]) and the group of automorphisms Aut( U M n) has cardinality 2 ℵ1 > ℵ 1 (see [1, ]). In particular, U M n has many external automorphisms in this situation. If (CH) fails, then things can look totally different, as the following result (due to the second author) on ultraproducts of the finite fields F p of prime order p shows. Theorem 1.1 ([5]). It is consistent with the axioms of ZFC that there is a nonprincipal ultrafilter F over the set P of primes such that the field F F p has no non-trivial automorphisms. Using this result, Simon Thomas and the first author proved the following result on ultraproducts of finite symmetric groups. Recall that for n 6 every automorphism of finite symmetric group Sym(n) is inner. In particular, every internal automorphism of the group U Sym(n) is inner. Theorem 1.2 ([3]). It is consistent with the axioms of ZFC that there is a nonprincipal ultrafilter U over ω such that all automorphisms of the group U Sym(n) are internal Mathematics Subject Classification. 03C20, 03E50, 20B30, 20E36. Key words and phrases. Ultraproducts, automorphisms. The research of the first author was supported by the Deutsche Forschungsgemeinschaft (Grant SCHI 484/4-1 and SFB 878). The research of the second author was supported by the German-Israeli Foundation for Scientific Research & Development Grant No /2007. Publication 982 on Shelah s list. 1

2 2 PHILIPP LÜCKE AND SAHARON SHELAH The work of this note is motivated by the following questions. Questions. (i) Is it consistent with the axioms of ZFC that for every non-principal ultrafilter F over the set P of primes, the field F F p has no non-trivial automorphisms? (ii) Is it consistent with the axioms of ZFC that for every non-principal ultrafilter U over ω, the group U Sym(n) has only inner automorphisms? The result of this note shows that a mild set-theoretic assumption implies the existence of ultraproducts of finite symmetric groups with external automorphisms. Theorem 1.3. Assume that the intersection of less than continuum-many dense open subsets of Cantor Space ω 2 is non-empty. If L is a finite first-order language and M n n < ω is a sequence of finite L-models containing models of arbitrarily large finite cardinality, then there is a non-principal ultrafilter U over ω such that U M n is infinite and has an external automorphism. The above assumption is known as Martin s Axiom for Cohen-Forcing and it is implied both by the continuum hypothesis and Martin s Axiom. 2. Extending Partial Elementary Maps From now on, we let L denote a finite first-order language and M n n < ω denote a sequence of finite L-models. We may also assume that M n < M n+1 holds for all n < ω. In addition, we fix the following objects. An enumeration c α 0 < α < 2 ℵ0 of all subsets of ω. An enumeration σ α 0 < α < 2 ℵ0 of all sequences in n<ω Aut(M n). An enumeration x α 0 < α < 2 ℵ0 of all sequences in n<ω M n. A regular cardinal θ such that H θ = {x tc(x) < θ} contains all relevant objects. Our aim is to inductively construct a non-principal ultrafilter U over ω and a function f : n<ω M n n<ω M n such that there is an external automorphism f of U M n with f([ x] U ) = [ y] U for all x, y n<ω M n with f( x) = y. We will approximate both U and f by their intersections with certain elementary submodels of H θ of cardinality less than 2 ℵ0. The following definition captures this idea. Definition 2.1. We call a triple U 0, N 0, f 0 adequate if it satisfies the following properties. (1) U 0 is a non-principal ultrafilter over ω. (2) N 0 is an elementary submodel of H θ of cardinality less than 2 ℵ0 containing f 0, U 0 and the sequence M n n < ω. (3) f 0 : n<ω M part n n<ω M n is a partial function and there is a partial elementary map f 0 : part U 0 M n U 0 M n with the following properties. (a) dom(f 0 ) = { x n<ω M n [ x] U0 dom( f 0 )}. (b) If x dom(f 0 ) with f 0 ( x) = y, then [ x] U0 N and f 0 ([ x] U0 ) = [ y] U0. Given adequate triples U 0, N 0, f 0 and U 1, N 1, f 1, we define U 0, N 0, f 0 U 1, N 1, f 1 to hold if N 0 U 0 U 1, N 0 N 1 and f 0 N 0 f 1.

3 EXTERNAL AUTOMORPHISMS OF ULTRAPRODUCTS OF FINITE MODELS 3 The idea behind this definition is that the submodel N 0 fixes subsets of U 0 and f 0 that will be contained in the final objects U and f (in the sense that N 0 U 0 U and f 0 N 0 f will hold). Note that f 0 might be the empty function in the above definition. In particular, there always exists an adequate triple. Moreover, if U 0, N 0, f 0 is an adequate triple, then the partial map f 0 is uniquely determined. Given U 0 and N 0 as in Definition 2.1, we define M U0,N 0 to be the unique L- substructure of U 0 M n with domain N 0 U 0 M n. We list some basic properties. Lemma 2.2. (1) Let U and U 0 be non-principal ultrafilters over ω and N 0 be an elementary submodel of H θ with U 0 N 0 and N 0 U 0 U. Then the map i : M U0,N 0 M n ; [ x] U0 [ x] U U is an elementary embedding. (2) Given an adequate triple U 0, N 0, f 0, we have f part 0 : M U0,N 0 M U0,N 0 and this is a partial elementary map. (3) If U 0, N 0, f 0 U 1, N 1, f 1 holds for two adequate triples, then the induced map i : M U0,N 0 M U1,N 1 is an elementary embedding with i f 0 = ( f 1 i) dom( f 0 ). Proof. Let ϕ ϕ(v 0,..., ϕ k 1 ) be an L-formula and x 0,..., x k 1 N 0 n<ω M n. By elementarity, we have {n < ω M n = ϕ( x 0 (n),..., x k 1 (n))} N 0 and the assumption N 0 U 0 U allows us to run an easy induction on the complexity of the formula ϕ to show M U0,N 0 = ϕ([ x 0 ] U0,..., [ x k 1 ] U0 ) U 0 M n = ϕ([ x 0 ] U0,..., [ x k 1 ] U0 ) {n < ω M n = ϕ( x 0 (n),..., x k 1 (n))} N 0 U 0 U U M n = ϕ([ x 0 ] U,..., [ x k 1 ] U ) The other statements follow directly from the above computations and the definitions of M U,N and f. Proposition 2.3. Suppose U 0, X, f and p satisfy the following statements. (1) U 0 be a non-principal ultrafilter over ω and X H θ with X < 2 ℵ0. (2) p : n<ω M part n n<ω M n is a partial function. (3) f : part U 0 M n U 0 M n is a partial elementary map with dom(f ) < 2 ℵ0, {[ x] U0 x dom(p)} dom(f ) and f ([ x] U0 ) = [ y] U0 for all x dom(p) with p( x) = y. Then there is an adequate triple U 0, N 0, f 0 such that p f 0, f = f 0, X N 0 and N 0 = X + dom(f ) + ℵ 0. Proof. We can find a partial function f 0 : n<ω M part n n<ω M n with p f 0, dom(f 0 ) = { x [ x] U0 dom(f )} and f ([ x] U0 ) = [ y] U0 for all x dom(f 0 ) with f 0 ( x) = y. Let D n<ω M n be a complete set of representatives for elements in dom(f ) with D = dom(f ). Finally, pick an elementary submodel N 0 of H θ with D X {f 0, U 0, M n n < ω } N 0 and N 0 = D + X + ℵ 0.

4 4 PHILIPP LÜCKE AND SAHARON SHELAH Lemma 2.4. Let U 0, N 0, f 0 be an adequate triple and x H θ. There is an adequate triple U 0, N 1, f 1 with the following properties. (1) U 0, N 0, f 0 U 0, N 1, f 1, N 1 = N 0 and x N. (2) dom( f 1 ) is a definably closed subset of M U0,N 1. Proof. There is a unique extension f part : M U0,N 0 M U0,N 0 of f0 to a partial elementary map whose domain is the definable closure of dom( f 0 ). Clearly, dom(f ) is definably closed in U 0 M n, dom( f) dom( f 0 ) + ℵ 0 N 0 < 2 ℵ0 and we can apply Proposition 2.3 with U 0, N 0 {x}, f and f 0 N 0 to find an adequate triple U 0, N 1, f 1 with the above properties. In particular, f = f and dom(f ) M U0,N 0 M U0,N 1. By the first part of Lemma 2.2, M U0,N 0 is an elementary substructures of M U0,N 1 and this shows that dom( f) = dom(f ) is definably closed in M U0,N 1. We are now ready to state and prove the Main Lemma needed in the proof of Theorem 1.3. This lemma allows us to enlarge the domain of our partial functions and make sure that the final automorphism will be external. Lemma 2.5. Assume that the intersection of less than continuum-many dense open subsets of Cantor Space ω 2 is non-empty. Let U 0, N 0, f 0 be an adequate triple with dom( f 0 ) definably closed in M U0,N 0, w, z N 0 n<ω M n with z / dom(f 0 ). There is an adequate triple U 1, N 1, f 1 with the following properties. (1) U 0, N 0, f 0 U 1, N 1, f 1 and N 1 = N 0. (2) z dom(f 1 ) and f 1 ([ z] U1 ) [ w] U1. Proof. We write z = [ z] U0 and M = M U0,N 0. For each n < ω, we equip the model M n with the discrete topology and the product K = i<2 ( n<ω M n) with the corresponding product topology. This means K is a perfect non-empty, compact metrizable, zero-dimensional topological space and homeomorphic to Cantor Space ω 2 (see [2, Theorem 7.4]). Define T to be the set of all tuples A, m, ϕ, x 0,..., x k 1 with the following properties. (i) A N 0 U 0, m < ω and x 0,..., x k 1 dom(f 0 ) N 0. (ii) ϕ ϕ(u 0, u 1, v 0,...,v k 1 ) is an L-formula and M = ( a, b) ϕ(a, b, [ x 0 ] U0,..., [ x k 1 ] U0 ). Given t = A, m, ϕ, x 0,..., x k 1 T with f 0 ( x i ) = y i, the above definition implies that the set D t = { a, b K ( n A \ m) M n = ϕ( a(n), b(n), y 0 (n),..., y k 1 (n))} is a dense and open subset of K. The set T has cardinality less than 2 ℵ0 and our assumptions imply that there is a pair a, b t T D t. Next, we define U to be the set consisting of triples u = A, ϕ, y 0,..., y k 1 such that A U 0 N 0, y 0,..., y k 1 N 0 n<ω M n and ϕ ϕ(v, x 0,..., x k 1 ) is an element of tp M ( z/ dom( f 0 )) with f 0 ( x i ) = [ y i ] U0. Given such an u U, we define X u = {n A M n = [ a (n) b (n) ϕ( a (n), y 0 (n),..., y k 1 (n)) ϕ( b (n), y 0 (n),..., y k 1 (n))]} and D = {X u u U}. Clearly, D is closed under taking finite intersections.

5 EXTERNAL AUTOMORPHISMS OF ULTRAPRODUCTS OF FINITE MODELS 5 Assume, toward a contradiction, that there is an u = A, ϕ, y 0,..., y k 1 U as above and m < ω with X u m. By assumptions, z / dom( f 0 ) and dom( f 0 ) is a definably closed subset of M. This shows M = ( a) ψ( z, a, x 0,..., x k 1 ) with ψ(u 0, u 1, v) [u 0 u 1 ϕ(u 0, v) ϕ(u 1, v)]. Given x 0,..., x k 1 N 0 n<ω M n with x i = [ x i ] U0, our assumption implies that the tuple t = A, m, ψ, x 0,..., x k 1 is an element of T. Since a, b D t, we can find a natural number n A\m with M n = ψ( a (n), b (n), y 0 (n),..., y k 1 (n)) and hence n X u \ m, a contradiction. This argument shows that there is a non-principal ultrafilter U 1 over ω with D U 1. For all A N 0 U 0, we have {n A a (n) b (n)} U 1. We can conclude N 0 U 0 U 1 and [ a ] U1 [ b ] U1. Therefore, we may assume [ a ] U1 [ w] U1. Let i : M U0,N 0 U 1 M n be the elementary embedding given by Lemma 2.2. Suppose x 0,..., x k 1 N 0 dom(f 0 ) and U 1 M n = ϕ([ z] U1, [ x 0 ] U1,...,[ x k 1 ] U1 ) with f 0 ( x i ) = y i. This means u = ω, ϕ(v, [ x 0 ] U0,...,[ x k 1 ] U0 ), y 0,..., y k 1 U and hence X u D U 1. In particular, {n < ω M n = ϕ( a (n), y 0 (n),..., y k 1 (n))} U 1 and therefore U 1 M n = ϕ([ a ] U1, [ y 0 ] U1,...,[ y k 1 ] U1 ). This implication shows that there is a partial elementary map f : part U 1 M n U 1 M n such that dom(f ) = i [ dom( f 0 ) { z} ], f ([ z] U1 ) = [ a ] U1 and f ([ x] U1 ) = [ y] U1 for all x N 0 dom(f 0 ) with f 0 ( x) = y. We can apply Proposition 2.3 with U 1, N 0, f and f 0 N 0 to obtain the desired adequate triple U 1, N 1, f 1. Corollary 2.6. Assume that the intersection of less than continuum-many dense open subsets of Cantor Space ω 2 is non-empty. Let U 0, N 0, f 0 be an adequate triple such that dom( f 0 ) is definably closed in M U0,N 0 and z N 0 n<ω M n. Then there is an adequate triple U 1, N 1, f 1 such that U 0, N 0, f 0 U 1, N 1, f 1, N 1 = N 0 and [ z] U1 ran( f 1 ) holds. Proof. We may assume [ z] U0 / ran( f 0 ). By the elementarity of f0, ran( f 0 ) is 1 definably closed in M U0,N 0. We apply Proposition 2.3 with U 0, N 0, f 0 and p = 1 to obtain an adequate triple U 0, M 0, g 0 with ḡ 0 = f 0, N 0 M 0 and M 0 = N 0. In particular, z / dom(g 0 ) and dom(ḡ 0 ) is definably closed in M U0,M 0. Next, we use Lemma 2.5 with U 0, M 0, g 0 and z to find an adequate triple U 1, M 1, g 1 with U 0, M 0, g 0 U 1, M 1, g 1, M 1 = N 0 and z dom(g 1 ). Let x N 0 dom(f 0 ) M 0 with f 0 ( x) = y. Then y M 0 dom(g 0 ) and therefore g 0 ( y) = x for some x M 0 with [ x] U0 = ḡ 0 ([ y] U0 ) = [ x ] U0. We can conclude {n < ω x(n) = x (n)} M 0 U 0 U 1 and ḡ1 1 ([ x] U 1 ) = [ y] U1. This allows us to use Proposition 2.3 with U 1, M 1, ḡ1 1 and f 0 to obtain the desired adequate triple U 1, N 1, f Approximations of External Automorphisms The next definition and the following lemma show how external automorphisms are constructed in a routine way from ascending sequences of adequate triples. Definition 3.1. A sequence U α, N α, f α, z α α < 2 ℵ0 is an approximation of an external automorphism if the following statements hold true for all α < β < 2 ℵ0.

6 6 PHILIPP LÜCKE AND SAHARON SHELAH (1) The triple U α, N α, f α is adequate and N α has cardinality α + N 0. (2) dom( f β ) is definably closed in M β = M U β,n β. (3) U α, N α, f α U β, N β, f β and c β, x β, σ β N β. (4) z β N β dom(f β+1 ) with z β / dom(f β ) and if w(n) = σ β (n)( z(n)) for all n < ω, then f β+1 ([ z β ] Uβ+1 ) [ w] Uβ+1. (5) [ x β ] Uβ+1 ran( f β+1 ) and if x β / dom(f β ), then x β = z β. Lemma 3.2. Let U α, N α, f α, z α α < 2 ℵ0 be an approximation of an external automorphism. Define U = α<2 ℵ 0 N α U α and f = α<2 ℵ 0 f α N α. (1) U is a non-principal ultrafilter over ω and f : n<ω M n n<ω M n is a total function. (2) There is an external automorphism f of U M n with f([ x] U ) = [ y] U for all x n<ω M n with f( x) = y. Proof. Our construction directly implies that U is a filter and non-principal. It is an ultrafilter, because every c ω is of the form c = c α N α for some 0 < α < 2 ℵ0. Given x n<ω M n, there is an 0 < α < 2 ℵ0 with x = x α N α. If x dom(f α ), then f α ( x) N α, f α ( x), x f α N α f and x dom(f). Conversely, if x / dom(f α ), then x dom(f α+1 ) N α and we can conclude x dom(f) as above. This shows that f is a total function. By Lemma 2.2, there are elementary embeddings i α : M α U M n that satisfy i α ([ x] Uα ) = [ x] U for all α < 2 ℵ0 and x N α n<ω M n and a directed system i α,β : M α M β α β < 2ℵ0 of elementary embeddings with i α = i β i α,β for all α β < 2 ℵ0. Moreover, we have i α,β f α = ( f β i α,β ) dom( f α ) for all α β < 2 ℵ0. This shows that there is a total elementary map f : U M n U M n with f([ x] U ) = [ y] U for all x n<ω M n with f( x) = y. Given x n<ω M n, there is an 0 < α < 2 ℵ0 with x = x α and this implies [ x] Uα+1 ran( f α+1 ). Since x N α+1, the above computations show that [ x] U ran( f) holds. This proves that f is surjective. Finally, assume, toward a contradiction, that f is an internal automorphism. Then there is an 0 < α < 2 ℵ0 with f([ x] U ) = [ y] U for all x, y n<ω M n with y(n) = σ α (n)( x(n)) for all n < ω. By definition, fα+1 ([ z α ] Uα+1 ) [ w] Uα+1 with w(n) = σ α (n)( z α (n)) and this means f([ z α ] U ) [ w] U, a contradiction. If Martin s Axiom for Cohen-Forcing holds, then we can apply Lemma 2.5 and Corollary 2.6 continuum-many times to construct an approximation of an external automorphism. The next lemma completes the proof of Theorem 1.3. Lemma 3.3. Assume that the intersection of less than continuum-many dense open subsets of Cantor Space ω 2 is non-empty. If U, N, f is an adequate triple, then there is an approximation of an external automorphism U α, N α, f α, z α α < 2 ℵ0 with U 0, N 0, f 0 = U, N, f. Proof. We inductively construct the approximating sequence. Assume we already constructed U α, N α, f α, z α α < γ with 0 < γ < 2 ℵ0 such that the statements (1), (2) and (3) of Definition 3.1 hold for all α < β < γ and the statements (4) and (5) of Definition 3.1 hold for all 0 < β < γ with β + 1 < γ. If γ = β +1, then z β / dom(f β ), dom( f β ) is definably closed in M β and there is w N β n<ω M n with w(n) = σ β (n)( z β (n)) for all n < ω. In this situation, we

7 EXTERNAL AUTOMORPHISMS OF ULTRAPRODUCTS OF FINITE MODELS 7 can apply Lemma 2.5, Lemma 2.4 and Corollary 2.6 to produce an adequate triple U γ, N γ, f γ with the desired properties. Let γ be a limit ordinal and define D = α<γ N α U α. This collection of subsets is closed under finite intersections and contains only infinite sets. Let U γ be a non-principal ultrafilter over ω extending D. As in the proof of Lemma 3.2, there are elementary embeddings i α : M α U γ M n and i α,β : M α M β with i α = i β i α,β for all α β < γ. Lemma 2.2 shows that there is partial elementary map f : part U γ M n U γ M n with dom(f ) = {[ x] Uγ x α<γ dom(f α)} and i α f α = (f i α ) dom( f α ) for all α < γ. Apply Proposition 2.3 with U γ, α<γ N α, f and α<γ f α N α and Lemma 2.4, to obtain an adequate triple U γ, N γ, f γ with the desired properties. By our assumptions, U γ M n is an infinite model and [4] shows that it has cardinality 2 ℵ0. This gives us an y n<ω M n with [ y] Uγ / dom( f γ ) and therefore y / dom(f γ ). By elementarity, we can assume y N γ. If x γ / dom(f γ ), then we define z γ = x γ. Otherwise, we define z γ = y. References [1] C. C. Chang and H. J. Keisler. Model theory. North-Holland Publishing Co., Amsterdam, second edition, Studies in Logic and the Foundations of Mathematics, 73. [2] Alexander S. Kechris. Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics. Springer-Verlag, New York, [3] Philipp Lücke and Simon Thomas. Automorphism groups of ultraproducts of finite symmetric groups. Accepted for publication in Communications in Algebra. [4] Saharon Shelah. On the cardinality of ultraproduct of finite sets. J. Symbolic Logic, 35:83 84, [5] Saharon Shelah. Vive la différence. III. Israel J. Math., 166:61 96, Institut für Mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstr. 62, Münster, Germany address: philipp.luecke@uni-muenster.de Institute of Mathematics, The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, Israel, and Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ , USA address: shelah@math.huji.ac.il

GROUPS WITH UNBOUNDED POTENTIAL AUTOMORPHISM TOWER HEIGHTS

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